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Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

Chapter: Water Shipping on a Vessel in Head Waves

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Suggested Citation:"Water Shipping on a Vessel in Head Waves." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Water Shipping on a Vessel in Head Waves M. Wreck, O.M. Faltinsen2, M. Landrinii (tINSEAN, The Italian Ship Model Basin, Roma - Italy, Department of Marine Hydrodynamics - NTNU, Trondheim - Norway) ABSTRACT We present an investigation based on experimental and numerical studies for the bow-deck wetness in head-sea conditions of a stationary ship, with a blunt bow. Three- dimensional water-on-deck experiments have been car- ried out by focusing wave packets against the model of a ESSO Osaka tanker. The experiments give a funda- mental description of the dynamics of water shipping and provide some useful data for the development of three-dimensional numerical methods. In particular, the observations confirm the formation of a cavity entrap- ping air during the initial stages of the water shipping. Near the fore portion of the deck, the local ship geom- etry affects shape and complexity of the cavity evolu- tion, always characterized by free-surface breaking and dispersion of bubbles in the main water field. To deal with this flow conditions, we present a Domain Decom- position (DD) strategy to solve numerically the prob- lem. A boundary element method (BEM) for the outer field, where smooth though steep waves are present, and a Volume-of-Fluid (VOF) method for the Navier- Stokes equations in the breaking region have been cou- pled by the DD. The present approach represents a com- promise between efficiency, robustness and capability of capturing the fragmentation of the air-water inter- face, which occurs at several stages during the flow evo- lution. The method is applied by assuming the flow to be two dimensional in the longitudinal ship plane, but no limitations exist to extend it to three dimensions. The DD approach appeared rather promising, although com- parisons with reference solutions demostrate that nu- merical inaccuracies develop in the VOF solver so far adopted. INTRODUCTION Reducing or preventing the occurrence of water ship- ping is becoming an important issue for safety and op- eration of ships. Compact masses of water entering and flowing on the ship deck are a danger both for advanc- ing and for moored vessels, whatever the considered ship type is. The present activity deals with the water on deck in the context of stationary ships. This is an important issue for FPSOs. These are ships with blunt bow forms. A turret-mooring system supported by thrusters and a dynamic positioning system are typically used. The ob- jective is to keep the vessel closely to head-sea con- dition. Green-water FPSO accidents documented deck wetness in the bow region as well as from the ship sides, with damages for deck house and equipment. In partic- ular, the location of the deck house can vary. FPSOs working in the North Sea usually have the deck house with living quarters in the bow region. Our focus is on the bow region, but green water has actually occurred at many different locations along the ship. Recently different projects have been program- med andlor carried out for a deeper understanding of the influence of ship geometry and wave parameters on the occurrence and severity of this phenomenon. In general, both experimental and numerical instruments are used for this scope. Our ongoing activity is aimed to understanding the phenomenon and developing robust and efficient pre- diction tools to be used in design and certification. In recent studies, we focused our attention on a simpli- fied prototype problem, Greco et al. (2000~. The how has been assumed two-dimensional in the longitudinal

ship plane. The unsteady interaction between free sur- face and ship has been accounted for, while viscous and surface-tension effects have been neglected. The problem has been solved numerically through a panel method (BEM) with piecewise-linear shape functions both for geometry and for boundary data. The method was verified and validated through comparison both with reference solutions and with experimental results. With this instrument, a simplified parametric analysis was carried out and the role taken by some of the main ge- ometric and kinematic parameters involved in the wa- ter on deck was discussed. Fundamental features of the phenomenon have been highlighted. Both water on deck resembling the breaking of a dam and water ship- ping due to large scale plunging waves have been inves- tigated. Two-dimensional water-on-deck experiments analyzed in Greco (2001) identified the efficiency and the limits of the numerical approach developed. During the tests, a restrained nearly rectangular ship model was exposed to head-sea waves generated by a flap wave- maker in a small wave flume. We discovered that water on deck starts in the form of a very localized plunging wave that breaks onto the deck, close to the bow. At the impact, an air cavity is formed. The post-impact evolu- tion is not handled by the panel method. This could be accomplished by using suitable high-speed local solu- tions for short time scales, or by different methods able to handle large deformation of the free surface, possibly with multi-phase flows. On a larger time scale, the ex- periments show a global behavior of the water along the deck similar but not quantitatively the same as the flow generated by the breaking of a dam, eventually hitting the deck house. Through a proper Kutta-like condition at the edge of the deck, the BEM method is able to cap- ture both the initial plunging phase and the global be- havior of the masses of water invading the deck. With this approach the high pressures related to the initial im- pact of the water with the deck house can be accurately estimated. On the other hand, the BEM is not able to handle post-impact phases, characterized by fragmenta- tion of the free surface. This occurs after the impact of the initial plunging wave with the deck and is associated with the collapse of the cavity. Similar situation char- acterizes the water field near the deck house in the late stages of the impact. During the water run-down along the wall a backward water overturning is observed, im- pacting with the under water flowing along the deck to- wards the deck house. Free-surface breaking and air en- trapment are observed. Both mentioned phenomena are potentially relevant from the structural point of view, since they could be associated with large loads on the structure. During the first stage, the loading due to the compression of the cavity is of main concern. Therefore its possible occurrence in the three-dimensional context is of practical importance. On the deck house, large loading occurs both when the water initially hits the structure and when the backward overturning water im- pacts and mixes with the under water flow, during the run-down phase. In this paper, the problem is analyzed experi- mentally and numerically. Three-dimensional water- shipping experiments have been performed to highlight the dynamics of the water flowing along the deck. On the numerical side, to deal with complex flow condi- tions, a Navier-Stokes solver with a Volume-of-Fluid (VOF) treatment of the interface is coupled via a Do- main Decomposition (DD) strategy with a Boundary Element Method (BEM), applied in the non-breaking region of the flow field. THREE-DIMENSIONAL WATER-ON-DECK EX- PERIMENTS Three-dimensional model experiments are ongoing to study water-on-deck phenomena for stationary ships. In the first part of our activity, we have focused our atten- tion on the flow evolution along the deck during the wa- ter shipping in head sea. A simplified analysis has been carried out by combining the flow visualizations with the wave elevation measurements around the used ship model, in the vicinity of the bow. Experimental Set-up The tests have been carried out at INSEAN facilities in a towing tank 220 m long, 9 m breadth and 3.6 m deep. The experimental set-up, as well as the incident wave parameters, have been decided by referring to PESO ships and their usual operational conditions. The FPSO water-on-deck accidents in North Sea documented that the most interesting wavelengths are of order of the ship length. In different cases, casualties occurred when the vessel was full loaded, with an effective freeboard defi- nitely smaller than the nominal value, Ersdal and Kvit- rud (2000~.

|\ "' ,~; r Figure 1: Body plan of the Esso Osaka ship model. A Esso Osaka ship model (see figure 1 ) has been used during the tests. The model has draft D ~ 0.284 m, length L ~ 4.44 m and beam B ~ 0.74 m. Since the ship model is restrained from oscillating during the experiments, and we wanted to have realistic heights of water relative to the deck for representative design con- dition, the upper portion of the bow has been modified by reducing the freeboard to f = 0.064 m (see lateral view in top plot of figure 2~. This means, D/L ~ 0.064, B/L ~ 0.166 and f /L ~ 0.015. The tank dimensions and the scale of the experiment ensure that the tests re- produce water-on-deck casualties in deep and open wa- ters. The ship bow has a conventional bulb, but this is likely to be not relevant for water shipping (ci i.e. Greco 2001~. Although the bow is quite blunt, at the deck level the bow sides form an angle of about 100° at the ship centerline. This is because the above water portion of the bow has been realized in a transparent material difficult to shape. This angle could affect the features of the initial stages of the water on deck. Also the deck has been realized in Plexiglas to permit visu- alizations of the water run-up along the bow and of the water front along the deck during the water shipping. This is achieved by using a video camera in combina- tion with a mirror placed under the ship deck (see left center plot of figure 2~. The flow evolution was moni- tored through a black/white video camera with a frame rate of 30 Hz. The video camera was placed inside the ship model and directed towards a mirror parallel to the ship deck and located onto the internal bottom of the model, as shown in the bottom sketch of figure 2. In this way, bottom views of the deck have been ob- tained. For quantitative analyses, the false prospective due to the mirror presence has to be filtered out. This was done by post-processing after the video camera has been calibrated. An additional color camera, with sam- pling frequency of 25 Hz, was utilized to have captain s~r~so~- rn 10 wet r s10OS7 0 s4 0 slO sllOs80 sl5° s20 sl20s90 s60 s30 camera _,_.,-,:z ' ~ i,. . I . . I . ~ ~ I ~ r ~ . /, ~ ~ mirror Figure 2: Experimental set-up: Lateral view (top), top view (left center), sketch with the wave sensors used in the experiments from top view (right center), and sketch of the camera-mirror combined system from lat- eral view (bottom). and lateral views of the water shipping phenomenon. A vertical wall in Plexiglas, perpendicular to the ship center plane, has been introduced along the deck, at a distance of 0.53 m from the bow (see right-center plot of figure 2), mainly to protect the video cameras used during the experiments. In the real case the water shipping may be due either to a single event, or to a summa of a small number of events related to wave groups approaching the ship. In the latter case, the first event is not necessarily the most severe one. In the present tests, the water on deck is studied as a single event. In this way, we also avoid effects due to wave reflections from the tank walls. The incident waves are wave packets. The used technique is based on the focusing of the wave energy by linearly decreasing the wavemaker frequency and thus increas- ing the group velocity of the generated waves leading

to focusing of the dispersive waves. In this context the energetic wave spectrum, the time instant for the focus- ing occurrence, and the focusing point along the tank, represent input data. In our case each wave packet is characterized by an identical amplitude a for all the generated wave components (see the amplitude spec- trum in left plot of figure 3~. A wave-steepness param- eter ha can be defined for the wave packet, k being the wavenumber associated with the mean frequency fc in the spectrum. In the tests, both fc and the frequency bandwidth Af are kept fixed and equal to 0.6 Hz and 0.4 Hz, respectively. This gives ~ = 2~/k = 4.33 m ~ 0.98L as wavelength related to fc, while the shortest and the longest wavelengths in each wave packet corre- spond to ~ 0.5 and to ~ 2.6 the ship length L, respec- tively. The wave steepness ha has been varied between 0.125 and 0.25. The focusing point has been chosen almost at the bow location (about 85 meters from the wavemaker). The typical time history of the wave ele- vation at this location without the ship model present is given in right plot of figure 3. The incident disturbances at 0~ . O 4J \/ Am/ · ~ W ~ t Figure 3: Left: example of amplitude spectrum associ- ated with the wave packets used the tests. Right: exam- ple of time history of the wave elevation at the location along the tank where the focusing phenomenon occurs, without the ship model present. are two-dimensional but the wave field in the vicinity of the ship will be three-dimensional due to the scat- tering of the incident waves. Sixteen capacitive wave sensors have been located around the ship model (see right bottom sketch of figure 2) to measure the wave el- evation and to quantify the effects due to the ship pres- ence. Each probe has a diameter of 0.5 mm to reduce as much as possible the disturbance to the flow field. Here, our attention is focused on the flow developing onto the deck during the water shipping and the wave scattering due to the ship is not discussed. General features of the water shipping The flow along the deck Two-dimensional water-on- deck experiments discussed in Greco (2001 ) have shown that the water shipping starts in the form of a wave plunging and hitting the deck near the bow of the ship model (see figure 41. A cavity entrapping air appears Figure 4: 2D water-on-deck experiments: plunging wave phase characterizing the stages of the water ship- ping. Time increases from left to right. and it is stretched downstream by the main flow along the deck and by the weight of new water entering the deck. Eventually the cavity collapses. The three di- mensional experiments confirmed the occurrence of a wave plunging phase at the beginning of the water on deck. The top plot in figure 5 shows a side view of the water shipping at about 0.12 s after the freeboard ex- ceedance. The incident waves propagate from left to right, and the interaction with the restrained ship model determines the bow deck wetness. In the case shown, the wave packet has a steepness ha = 0.15 and central wavelength ~ = 4.33 m (case b). The water shipping starts first from the fore portion of the bow, where the wave elevation exceeds earlier the freeboard. Then the phenomenon developes along the bow sides, with a non- uniform (decreasing) maximum freeboard exceedance going from the fore portion towards the superstructure. The water initially enters the deck in the form of a plung- ing wave hitting the deck near the bow sides. The arrow in the snapshot indicates the flow region characterized by a plunging wave. To better emphasize the plunging wave phase, the ship deck has been removed and the resulting flow is shown in the bottom plot of the same figure, for a time instant of about 0.16 s after the free- board exceedance by the water. A top-captain view of the water on deck in the same conditions is shown in figure 6. The horizontal line visible in the snapshots in- dicates the location of the vertical wall along the deck. The time increases from left to right and from top to bottom, with interval of 0.08 s between two snapshots. In the pictures, the yellow color is the internal color of the ship model. The green color in the water is due to a mixture of natrium flourisenium powder placed in the upper-front portion of the bow and partially convected by the water flow during the run-up along the bow. The mixture has been used to make clearer the water ship- ping features. The water shipping starts first near the

~ ~ ~:~ - - -a i. ] ~ - - ~ - - Figure 5: 3D water-on-deck experiments: side view. The wave packet has ha = 0.150 and ~ = 4.33 m (case b). Top: snapshot of the water shipping at about 0.12 s after the freeboard exceedance. Bottom: snapshot of the water shipping at about 0.16 s after the freeboard exceedance in the case when the ship deck has been re- moved. Figure 6: Water on deck caused by a wave packet with ha = 0.150 and ~ = 4.33 m (case b), top-captain view. Time increases from left to right and from top to bottom. The time interval between two snapshots is ~ 0.08 s. terline, resulting in a tongue of water moving towards the superstructure. This one is clearly characterized by a smaller layer of water than the two lateral jets gen- erated from the impact. The latter leave the ship axis of symmetry in the form of water humps entrapping air. As a consequence of these reflected flows, the cen- tral tongue enlarges its width with time and is delimited by a non-smooth structure, likely characterized by mix- ing of water and air bubbles. At this stage, along the bow sides the two cavity arms have collapsed and two jet flows (one for each side) are originated towards the vertical wall, moving slower than the central tongue- shaped flow. The described flow evolution along the r: IF ship centerline, and proceeds progressively from there I on. Two non-uniform cavities are originated along the bow sides. The initial water-on-deck phase at the fore bow is not very clear from the snapshots. There, the plung- ing waves from the two sides of the bow interact with each other and with the flow entering the deck along the ship centerline. The water fronts coming from the two ship edges interact along the ship center plane, and are reflected outwards leaving laterally the deck. Air is entrapped and convected by the two flows. The charac- teristics and the collapse of the cavities are the result of the local flow features and of the local bow char- acteristics. The mentioned impact causes also an in- crease of the velocity of the flow along the ship cen- - ,\ 1 11 1 Figure 7: Sketch of the flow evolution during the water shipping, top-captain view. Time increases from left to right and from top to bottom. deck is qualitatively sketched in figure 7. The instan- taneous water fronts along the deck are better observed from the bottom view presented in figure 8. The time in-

creases from left to right and from top to bosom, with a time interval of about 0.33 s between two snapshots (the prospective error has not been corrected in these images). Figure 8: Water on deck caused by a wave packet with ha = 0.15 and ~ = 4.33 m (case b), bottom view. Time increases from left to right and from top to bottom. The time interval between two snapshots is ~ 0.033 s. The horizontal black lines visible in the snap- shots are the result of the reflection from the mirror of the lines drawn on a panel in front of the bow (the panel is indicated by the arrow in the top plot of figure 2~. From the pictures, near the fore portion of the deck, a circular-shaped structure is formed at the be- ginning of the water shipping, increasing as the time increases. In the second snapshot, this structure is char- actenzed by an inner-darker region, where the deck has been already wetted, and an outer-lighter ring, where air is present. At this instant two cavities are observed along the bow sides, they have been colored in yellow to make them more evident and are indicated by the ar- rows. These are not perfectly uniform, with thickness slowly reducing from the ship centerline on. At this stage, the impact of the water with the deck has already occurred. Two types of jet Rows have been originated along the deck? almost perpendicular to the bow sides and, respectively, entering and leaving the deck. This is similar to the observations in the two-dimensional ex- periments in Greco (2()01, see right plot of figure 41. The evolution of the water near the centerline needs additional comments. There, the flow is well three- dimensional. After the cavity collapse, a main jet to- war~is the superstructure is Unnerved, accompanied by two lateral flows leaving the deck (C,76 second plot in the figure). The former is faster than the water fronts along the bow sides entering the deck, as well as the two lateral flows are faster than the water fronts along the bow sides leaving the deck (ct third plot in the figure and the enlarged view given in figure 9~. The region of ~ . ,,~r~,;,,, . a: ,., ,:, . Figure 9: Water on deck caused by a wave packet with ka = 0.10 and ~ = 4.33 m (case b), bottom view. En- larged view of the flow region near the ship centerline at the time instant show n in third plot of figure 8. the deck interested by the evolution of the cavity struc- tures until their collapse is well delimited by two lines of water (one at each bow side). The rest of the deck is gradually wetted by the water flowing towards the vertical wall. The collapse of the cavities leads to bub- bles, first in the form of organized structures and then as individual bubbles dispersed within the main bulk of water. These are larger near the centerline. The volume of air initially entrapped is larger near the ship center- line and reduces moving along the edges of the deck. Consistently, we observed larger bubbles near the ship centerline. The flow of water towards Me vertical wall does not appear completely smooth. Wavy structures with relatively short wave length (order of one centime- ter) are observed. Thes ripples are probably caused by surface tension effects, relevant at the beginning of the water shipping due to the high curvatures involved. The global shape of the water fronts developing along the deck is initially similar to a straight line, but it diverges progressively from this behavior as the time increases.

At the same time the water fronts become less smooth. Figure 10: Water on deck caused by a wave packet with ha = 0.150 and ~ = 4.33 m (case b). Left: lateral view of the water shipping before the water impact with the vertical wall along the deck. Right: top-captain view of the water shipping during the water run-up along the vertical wall. The water impact with the superstructure The im- pact of the water with the vertical wall along the deck occurs generally in a non-uniform way. This is shown in figure 10 through a lateral view (left) of water shipping just before the impact, and a top-captain view (right) of the water shipping during the water run-up along the structure, once the impact has occurred. The flow run up along the wall is eventually reversed due to the gravity action, and a backward water overturning is ob- served (see top plots of figure 11~. The same phenome- L__11~ it: - ~ i ~ BY 1 __ ~ _ I _ -God ~ ~~ J :~_d _ _—_ _ _ ___ ~r'~ ~ _ - - - - - _~ l _ Figure 11: Backward plunging wave formed after the maximum run up has been reached. Top: 3D wa- ter-on-deck experiments. Bottom: 2D water-on-deck experiments. Time increases from left to right. non has also been observed during the two-dimensional experiments in Greco (2001, see bottom plots in the same figure) and pressure measurements along the ver- tical wall showed an increase of the structural loads due to the plunging formation and its final impact with the under water flowing towards the superstructure. It is reasonable that the same happens in the three dimen- sional case. The water-off-deck phenomenon The water how gen- erated by the reflection from the vertical wall causes eventually a water-off-deck phenomenon (see left plot of figure 12) modifying the flow field conditions around the ship bow with respect to the diffracted wave field without water-on-deck occurrence. As an example, at the location of sensor 7 (indicated by the arrow in the left plot of figure 12) a primary peak of the wave eleva- tion corresponding to the water-on-deck occurrence is followed by a sharp secondary peak (see right plot of figure 12) associated with the disturbance of the water leaving the deck. Therefore, in case of repeated water Figure 12: Water on deck caused by a wave packet with ka = 0.150 and ~ = 4.33 m (case b). Left: side view during the water-off-deck phenomenon. Right: time history of the wave elevation at the wave sensor 7 (indi- cated by the white arrow in the left photo). At the time instant of the left snapshot a sharp secondary peak in the measured wave elevation is observed, as indicated by the connecting arrow. shipping, the second water on deck could be strongly affected by the modifications in the wave field around the ship bow due to the water-off-deck phenomenon as- sociated with the previous event. Steepness influence on the water shipping Figure 13 shows snapshots of the water along the deck after the collapse of the cavities associated with water- on-deck events due to wave packets with ~ = 4.33 m and ha = 0.125, 0.15, 0.175, 0.2 and 0.225, respec- tively (cases a-e). The non-dimensional time instants are not the same, and correspond to /\7wO~ = (t—twos) /~/(H—f)/g ~ 0.95, 0.83, 0.87, 0.79 and 0.74, for increasing values of the considered steepnesses. Here,

~ - ~ - - ~ — - - ~ ~ - - ~ - ~ Figure 13: Steepness influence on the initial cav- ity. Water on deck caused by wave packets with ~ = 4.33 m and ka = 0.125, 0.150, 0.175, 0.200 and 0.225 (case a,.., case e, respectively), bottom view. r i\~ = (t—twod)/~\/(H—;)|g,whereH = 2a end twod is the time instant when the water shipping started. The spatial variables are made non-dimensional by (H - f ). The prospective error has not been corrected. twos is the time instant when the water shipping started. In the case with the smallest incident wave steepness, the region of the deck near the fore portion of the bow appears almost completely wetted, differently for the other cases the ring-shaped regions near the bow apex, identifying the collapse of the two cavities at the fore bow, and the related lateral flows, are visible. As the steepness increases, the bubbly structures associated with the two cavities become less uniform along the ship sides, and thicker near the ship centerline. Snapshots of the water along the deck, at the non dimen- sional time instant i\rWO~ ~ 1.6 after the starting of the water shipping and for the same cases, are shown in fig- ure 14. As we can see, the bubbly structures subsequent to the cavities collapse (especially near the ship cen- terline) increase with the steepness. Also, the portions of the deck interested by the evolution of two cavity arms become larger as steepness increases. These can- not even be clearly detected in the case with the small- est steepness (case a). The water fronts during the water shipping have been extracted by the video images and are presented in figure 15. In the plots, the water en- ters the deck from the bottom and the fore portion of the ship deck is represented by the straight lines form- ing an angle of about hundred degrees. The time in- terval between two water-front configurations is equal to the frame rate of the video camera (1/30 s). As the wave steepness increases, the water fronts become less Figure 14: Steepness influence on the water front evo- lution. The water-on-deck is caused by wave packets with ~ = 4.33 m and ha = 0.125, 0.150, 0.175, 0.200 and 0.225 (cases a-e, respectively), bottom view. The snapshots correspond to the time instant t ~ 1.6~/(H—f)/g from the starting of the wa- ter shipping (H = 2a). The spatial variables are made non-dimensional by (H—f). The prospective error has not been corrected. smooth and with a more marked interaction of the cen- tral water flow with the lateral ones. In the last plot (right bottom) of the figure, the estimate of the front ve- locity along the ship centerline is given. This has been obtained as ratio between the distance covered by the water front and the related time interval. The different cases are associated with non-dimensional front veloci- ties of the same order of magnitude. Case a, for which a longer non-dimensional time evolution is available, shows a water front velocity slowly increasing with a rate of change reducing with time. This suggests an in- creasing shallow-water behavior of the water. A similar behavior is also shown by case b. Differently the other cases (cases c-e) show a relatively large change with time of the water front velocities. The data available for these cases refer to a non-dimensional time inter- val smaller than for the two less steep cases. The large rate of change in the front velocity implies that shallow- water conditions are not yet reached by the flow along the deck, and dispersion effects still matter. TWO-DIMENSIONAL NUMERICAL STUDIES Two- and three-dimensional tests evidenced breaking and fragmentation of the free-surface, and air entrap- ment during initial and late stages of the water ship- ping. These phenomena are relevant for the resulting

and Valli 1999), where the flow field is divided into sub-domains, each one modeled by a different numer- ical method. In this way, the advantages of the BEM method can be combined with the capabilities of a field method. In particular, the latter will be used to describe the flow evolution in proximity of the ship bow and on the ship deck. The rest of the fluid domain will be ac- curately described by the BEM, that is more efficient and accurate in dealing with non-breaking free-surface flows than field methods. 200 1X 200 1X 300 _ a 1= 0 1 300 ; _~ WIt case ~ Urn Ann 100 a,~|t ~ \1// case b . -~_ \ / case c .... ~ . , .. I .... ~ ... I^'~ fit \- / can 0 _ 200 1= :~ 0 \ /ca'cd O lCO 200 30O x (mill) 0 ease a · ease b .~ . v g(H-~) as / . O- ease c card _ . . ° ~ ~ Ax~/(HJ) 3 Figure 15: Case a to case e: water front propagation along the deck (bottom view), time interval equal to the rate of the video camera (1/30 s). The dotted lines in the plots indicate the lines tangent to the bow sides and the ship centerline. Right-bottom plot: water front velocity along the ship centerline as a function of the distance of the water front from the most fore bow point. H = 2a. structural loads and should be correctly treated by any prediction tool. As shown in Greco (2001), a BEM approach is able to detect and quantitatively capture several features of water shipping but cannot handle the long-time evo- lution, when free-surface breaking and two-phase flows are observed. To the purpose, a field method with a suitable treatment of the air-water interface could be used. On the other hand, field methods are more expen- sive in terms of memory and CPU-time requirements. Moreover, the present experiments evidenced relevant three-dimensional effects during the entire water ship- ping, both around the ship and on the deck. The use of field methods may become prohibitive when dealing with the three-dimensional problem. As a compromise between efficiency, robustness and ability to handle the air-water interface dynamics through the entire evolution, we decided to develop a Domain Decomposition (DD) strategy (see Quarteroni In the whole domain the flow is assumed incom- pressible, while only in the BEM sub-domain it is con- sidered inviscid and irrotational. Surface tension is ne- glected everywhere. A free-slip condition along rigid boundaries and pressure continuity across the air-water interface are enforced. In the region studied with a field method, the problem is written in terms of primitive variables, ve- locity and pressure, and the numerical solution is ob- tained through a finite difference technique, on a stag- gered Cartesian grid. A two-step projection algorithm is used, where an auxiliary non-free divergence velocity is introduced. The pressure field is governed by the Pois- son equation and it is solved by an incomplete Cholesky conjugate gradient method. The free-surface is treated by a Volume-of-Fluid (VOF) technique, where the inter- face is reconstructed by means of a passive scalar field J(P, t) ~ LO, 1], representing the local water-volume fraction. Details for the numerical algorithm can be found in Hirt and Nichols (1981~. The BEM method and the related numerical details are extensively descri- bed in Greco (2001~. Within the DD approach, BEM and VOF regions are connected by a Transmission Domain portion (TD), where the two sub-domains exchange information. In this context, two different procedures have been ana- lyzed (Campana and Iafrati 2001~. In the first one, cp procedure a in the top sketch of figure 16, TD is a con- trol surface and represents a boundary portion of the BEM and VOF sub-domains. In this case, the velocity distribution calculated by the BEM is used as boundary condition for the field method. The latter provides the pressure distribution used, through the Bernoulli equa- tion, to update the velocity potential enforced along the transmission boundary. Here a Dirichlet boundary con- dition is enforced. In the procedure b in the bottom sketch of fig-

a) I TD ---- --. i , s 1 . : : : : : ~ : ::: ::: :: : :: 13: : . BEM region : : ::: : :: : VOF: region i: :~: ~ I ~ ~ ~ i: 1 :: ~ : - i:~:~ ::~ ::::: :::~::::: :~::: ::: ~:: I::::: ::: :: :::: ::::::::: :::: ::: :::: :::: :::: :: ::::: ::::::: :::::::::::: ::::::::::: ::::::: 1::::: :::: ::: :: :::: :::: :::: :::::::::: ::::: I: : ::: :: :: : :: ::: ::::: ::::::: · : ~ : : : - ~ :: A: : : : : : ::: ............ BEM boundary" VOF boundary hi) 1|, TD . _ _ rid : art:: r~ I ~: r— rid A: BEM region r~~ . ~ VOF boundary t ~ BEM boundary :: ~~VOF~:reg~on~ :~: Z . . :: : : :: : : : Figure 16: Definition of the transmission region within the Domain Decomposition approach according to pro- cedures a and b explained in the text. ure 16, the two sub-domains are partly overlapped, and TD is the domain portion belonging to both of them. In the computations, the VOF solution gives the normal velocity component enforced along the BEM boundary (Neumann boundary condition), while the BEM solu- tion gives the velocity distribution required by the field method along the VOF boundary. In the numerical implementation, the time step is governed by the stability constraints of the field method, more stringent than those requested by the free-surface evolution in the BEM domain, where a standard fourth- order Runge-Kutta method is adopted. Though the approach is rather general, at present only the two-dimensional version of the method has been implemented. Examples of application of the method are presented in the following. In particular, we men- tion that in the present case the emphasis is not on vis- cous effects, and, for instance, the grid resolution has not been chosen refined enough to capture boundary layer effects and, consistently, a free-slip condition is enforced along the solid boundaries. Dam-breaking problem As a preliminary study, the DD approach has been ap- plied to a dam-break problem. A vertical (rigid) wall is located downstream the initial dam position to mimic a deck-house impact event. The considered reservoir of water has a height h and a length 1. The DD results shown in the following have been obtained by using procedure a described above for the coupling between the two sub-domains. The BEM solver is used to simu- late the flow evolution up to a given horizontal distance from the dam. Downstream this vertical section, the problem is solved by the VOF method (see Figure 17~. The VOF region is rectangular shaped, and its boundary dam ~ ~W~W_~W~ BEM sub-domain ~ VOF sub-doma~n I TD ~ :~ Hi: ~: : ~ ~ ~ A:: A: ::: ~ A:: ~ :::: ::: :::: ::::::: ::::: :: ~ ~ ~ ~ ~ ~ - ~ : ~ - ~ ~ . . - ~ : ::: ::: :::::: :: :::::::::: :::::: A: ~ : ~ .: . . : ::: ~ A::: :::~:: ~ :~ : : A:: ~ . : : ::: :: _ ~ _ . ~~\~\~\~ ~ . ~ x Figure 17: Definition of BEM and VOF regions for the problem of dam breaking followed by the impact with a vertical wall. is made by TD (left side), the 'deck' (bottom side), a vertical wall (right side) and the top boundary, which is modeled as a rigid boundary. The velocity computed by the BEM is enforced along the transmission boundary, while a free-slip condition is enforced along the remain- ing portions of the boundary. Once the dam has been broken, the water flows along the initially dry deck. The field method is switched on at t = t*, after the water entered the VOF sub-domain. For t < t*, the BEM is used to solve the whole problem. For t > t* on, the BEM sub-domain is limited in the rightward extent by the transmission boundary. A Lagrangian algorithm is used in the BEM sub-domain, therefore, during the evo- lution, those free-surface points entering the VOF sub- domain are eliminated. Occasionally, new free-surface points are introduced in the BEM domain by using cubic- spline interpolation procedures. Figure 18 gives snapshots of the free surface dur-

o DD sol. —BEM sol. x/n o x/n x/n . . z o DD sol. , o DD sol. —BEM sol. —BEM sol. . Omission boundary . = . —Am_ . it' 2 0 DD sol. ~ ' x~ 2 O DD sol. i · xl7 —BEM sol. —BEM sol. O . . O z o DD sol. ~ ' x~ ~ o DD sol. 2 ' me —BEM sol. —BEM sol. . , ~ e——e s e ~ ~ e ~ O l l O 2 o DD sol. ' x/h 2 O Dom. Decom.soL ' xAi —BEM sol. —BEM sol. ' zip . I' o 0 7 ~ X 0 2 ~ X'7 Figure 18: Dam-Breaking problem (I = 2h) and im- pact with a vertical wall at 3.366h far from the dam. Free surface configurations at t~/~ =2.2, 2.6, 3.6, 4.1, 4.6, 5.1, 5.6 and 6.2. Time increases from left to right and from top to bottom. DD results (circles) are compared with BEM results (lines). O DD sol. —BEM sol. a DD sol. —BEM sol. r/h 0 DD sol. — REM ~1 r/h o Dom. Decom. sot —BEM sol. r/h ing the flow evolution after the dam break. In this case I = 2h and the downstream wall is placed at 3.366h from the initial dam. The studied case is the same as discussed in Greco et al. (2000~. Results are presented for the DD case (circles) with transmission boundary located at 1.3h from the dam, and for the full BEM simulation (lines). In the figure, the nondimensional time t~/57; increases from left to right and from top to bottom. The overall agreement between the two re- sults is satisfactory. The initial impact and the water rise up along the vertical wall are well captured by the DD approach, although a flow instability seems to de- velop during the water run-up along the vertical wall (ct sixth plot of the figure). The results of the two methods start to diverge quantitatively in the late stages, during the water run down, when the backward plung- ing is formed, finally hitting the underlying water. In particular, in the DD solution, the impact phenomenon occurs earlier and closer to the vertical wall. The full BEM solution agrees quite well with that obtained by the Smoothed Particles Method (SPH), a Lagrangian meshless field method, Tulin and Landrini (2000), as the comparison for the two latest time in- stants confirms (ci figure 19, ct Colicchio et al. 2001~. Therefore, we consider the full BEM solution as our reference solution in verifying our DD algorithm. We have seen that the present DD results are not affected . SPH sol. —BEM sol. . SPH sol. —BEM sol. Figure 19: Dam-Breaking problem described in figure 18. Free surface configurations at t~7; ~ 5.6 (left) and 6.2 (right). BEM results (lines) are compared with SPH results (squares). by the coupling procedure adopted, and we are prone to believe that the accuracy of the field-method imple- mented has to be improved to better handle the entire evolution of the phenomenon. Water-on-deck problem The DD algorithm has also been applied to simulate the water entering on a ship structure with finite draft. As previously mentioned, in this case the VOF sub-domain is restricted to the deck region and a relatively small region of the flow in front of the bow (see sketch 20), the rest of the domain is studied by using the BEM solver. Also for this problem, we tested both the cou- pling procedures and discovered that only the procedure b gives reliable results for long-time evolutions. Specif- ically, from our studies, both the procedures a and b are applicable when the main flow travels from the BEM sub-domain towards the VOF sub-domain. When the main flow travels from the VOF sub-domain towards the BEM sub-domain (e.g. in the case of wave reflections by the ship bow as occurring during the water-on-deck problem) the coupling procedure a breaks down. =:= ~ :: transmission ~~ ~ :~ :~ : boundary - ~ VOF 5ub~domain ~"%, ',(, ........ BEM sub~domain _....... Figure 20: Definition of BEM and VOF regions for the problem of water on deck on a ship structure with finite draft. Therefore, the following results are based only on the coupling procedure b. Figure 21 gives snap- shots of the free surface during the initial stages of the

water shipping onto the deck of a nearly rectangular ship structure. The considered case corresponds to the two-dimensional water on deck experiments reported in Greco (2001, see figure 22 where the experimental re- sults are satisfactorily compared with the BEM results accounting for the surface tension effects). The DD re- 0.4 0.2 z/D 0.4 0.2 ED .q . 0.2 . ' · ~ _1 7 —BEM sol. . —BEM sol. . o DD sol. 0.2 . ° DD sol. -3.9 X/Z) .3.6 -3.9 AD -3.6 -3-9 X/D -3.6 . . ID ~ -_ 0.4 . —BEM sol. . —BEM sol. . o DD sol. 0.2 . ° DD sol. -3.9 x./D -3.6 . . _ ID ~ 0.4 0.2 \ r —BEM sol. o DD sol. fit _ 0 DD sol. -3 9 x/D -3.6 -3.9 x/D .3.6 Figure 21: Water-on-deck on a nearly rectangular ship structure: initial stages. Free surface configurations at t ~ 55.5, 55.7, 55.9, 56.0, 56.3 and 56.4 >/~7; (the initial time corresponds to the starting of the wavemaker motion). The time increases from left to right and from top to bottom. Domain-Decomposition results (sym- bols) are compared with full BEM results (solid lines). suits (symbols) are compared with the full BEM results (solid lines). The two approaches are in good agreement both during the water run-up along the bow and during the formation of the plunging wave, after the freeboard has been exceeded by the water. However, they progres- sively diverge when large variations of the free surface have to be handled. The results confirm the efficiency of the DD strategy but still suggest the need to improve the VOF treatment so far adopted. CONCLUSIONS Three-dimensional experiments have been presented to highlight the dynamics of the water shipping and the Figure 22: 2D water-on-deck experiments: initial plunging phase. The red lines are obtained numerically by the BEM method with included surface tension ef- fects. fluid flow evolution on the ship deck. In the experi- ments, use has been made of wave focusing to gener- ate a single water-shipping event. A restrained to move ESSO Osaka model has been adopted. Previous two- dimensional experiments have been confirmed in show- ing that water shipping starts with the fluid plunging onto the bow deck. At the impact, cavities of complex shape, entrapping air, are formed. The cavity evolu- tion is rather complicated and characterized by stretch- ing and, eventually, fragmentation, leading to flow. In more detail, after the freeboard is exceeded, the water fronts plunge onto the deck, forming two non- uniform cavities. These develop along the bow sides, with the maximum cross sectional area at the ship cen- terplane, decreasing along the bow edges. Soon after, the water fronts move inwards, and meet each other along the ship centerplane, giving rise to a vertical splash up, eventually reversing outwards. In the meanwhile, the cavities are stretched and finally broken into bub- bles of variable size. The details of this evolution depend on different factors related both to the incoming waves and to the ship bow shape. The influence of the incoming wave steepness on the cavity and on the water front propaga- tion has been analyzed. The importance of dispersion effects during the water shipping has been discussed. According to a simplified two-dimensional analysis car- ried out in Greco (2001), the loads on the deck associ- ated with the compressibility of the air entrapped into the cavity can be relevant for deck safety. At present, no pressure measurements have been performed to quan- tify such loads and the related effects on the structure in the three-dimensional case. However, the tests permit to identify the extent of the cavity region near the bow and the time scale of its collapse. Such information is also useful for the set-up of further dedicated tests to measure the pressure in areas where the cavity devel- ops. A second-step experiment is planned for two dif-

ferent bow shapes, with different bluntness, to perform a more systematic analysis of the three-dimensional ef- fects in the context of the water-on-deck phenomenon. As the numerical modeling of green-water phe- nomena is concerned, we have presented a Domain De- composition strategy to extend the investigations per- formed by the Boundary Element Method, Greco et al. (2000~. The underlying idea is to use a field method with a more general treatment of the free-surface only in domain regions where breaking and two-phase flow is observed, and applying the BEM in the rest of the fluid domain. In the present implementation, we adopted a VOF technique to handle the free surface in the field- method sub-domain. Examples given show the potentiality of the com- putational strategy, although a more refined VOF tech- nique has to be developed to increase the reliability of the whole method. ACKNOWLEDGEMENTS The INSEAN research activity has been supported by the Italian Ministero per le Infrastrutture ed i Trasporti through INSEAN Research Program 2000-02. The authors are indebted with Dr. C. Lugni (INSEAN) for his expertise and help in performing the three di- mensional experiments. REFERENCES Campana, E. and A. Iafrati. "Unsteady Free Surface Waves by Domain Decomposition Approach". Proc. of 16th Int. Workshop of Water Waves and Floating Bod- ies. Hiroshima, Japan, 2001. Colicchio, G., A. Colagrossi, M. Greco, and M. Lan- drini. "Free-surface Flow After a Dam break: A Com- parative Study". Proc. of 4th Numerical Towing Tank Symposium (NuTTS). Hamburg, Germany, 2001. Ersdal, G. and A. Kvitrud. "Green water on Norwe- gian production ships". Proc. 10th Int. Conf. Offshore and Polar Engg, ISOPE'2000. Seattle, 2000. Greco, M. A Two-dimensional Study of Green-Water Loading. Ph. D. thesis, Dept. Marine Hydrodynamics, NTNU, Trondheim, Norway, 2001. Greco, M., O. M. Faltinsen, and M. Landrini. "Basic studies of water on deck". Proc. of 23 rd Symp. on Naval Hydrod. National Academy Press, Washington DC, Vat de Reuil, France, 2000. Hirt, C. W. and B. D. Nichols. "Volume of fluid (VOF) method for the dynamics of free boundaries". J. of Computational Physics Vol. 39, pp. 201-225,1981. Quarteroni, A. and A. Valli. Domain Decomposition Methods for Partial Differential Equations. Oxford Sci- ence Publications, 1999. lblin, M. P. and M. Landrini. "Breaking waves in the ocean and around ships". Proc. of 23r~ Symp. on Naval Hydrod. National Academy Press, Val de Reuil, France, 2000.

DISCUSSION D.K.P. Yue Massachusetts Institute of Technology, USA The present paper is a very useful addition to the literature on this important problem. The presentation of three-dimensional data is of particular value. The authors' experimental observation that, unlike dam breaking, water on deck starts in the form of a localized plunging wave, is of special interest. In general, one would expect that the formation of a plunging wave during water entry onto the deck must depend on the speed of the oncoming wave crest and the geometry of the hull (and deck). If the speed of the wave crest is relatively small and/or the edge between the hull and deck is smoother), the present conclusions may not be valid. Did the authors investigate the effects of wave speed and hull-deck geometry? AUTHORS' REPLY The wave speed parameter has been varied during our three-dimensional experiments by varying the wave steepness (see the paper). In all the cases the plunging phase was observed. We also performed two-dimensional laboratory tests (see t13), and for the smallest wave amplitude we observed a gentler overtopping. In this case, surface-tension effects alter the particle trajectory reducing the entrapped cavity and helping a sort of blunter impact. However there is still a cavity formed. Rounded edges of the overdeck portion of the hull sides are rarely seen in ships, therefore, large scale experiments, and the actual geometry of ships suggest that air-cavity formation is the most frequent behaviour in practical cases. DISCUSSION M. Kashiwagi Kyushu University, Japan In the domain-decomposition method, we need to transfer the data (like pressure, normal velocity) from one domain to another domain. I suspect this procedure decreases the accuracy, and eventually the BEM might break down. Aren't there any problems associated with numerical instability and inaccuracy in the domain-decomposition method? AUTHORS' REPLY Domain Decomposition (DD) strategies have strong theoretical foundations which guarantee that, under suitable refinement, the solution converges to the solution of the single-domain problem (see t23~. There are no indications that the BEM method cannot be used within a DD approach. Wang et al., t3], in fact, adopted the DD strategy to speed up the solution of Boundary Integral Equations (BIE) in large domains and showed high accurate results. In zonal approaches, where DD is used to couple different solvers for different Boundary Value Problems, the coupling procedure between the sub-domains determines the accuracy and the stability of the solution. In the present paper, we have investigated two coupling strategies, indicated as a and b, and we found that procedure a has a poorer stability than procedure b. We cannot ascribe this behaviour just to the BEM but to the specific details of the coupling between the BIE and Navier-Stokes solvers. 1. Greco, M., "A Two-dimensional Study of Green-Water Loading", PhD Dissertation, MTA- rapport 2001-146, Dept. of Marine Hydrodynamics, NTNU, Trondheim, Norway, 2001. 2. Quarteroni, A., and Valli, A., Domain Decomposition Methods for Partial Differential Equations Oxford Science Publications 1999. , , 3. Wang, P., Yao, Y., and Tulin, M. P., CCAn efficient numerical wave tank for nonlinear water waves, based on the multi--subdomain approach with teem", Int. Journal for Numer. Meth. in Fluids, Vol. 20, 1995.

O,04 0,03 0,02 0,01 0,00 -0,01 -0,02 -0,03 -0,04 0,10 0,08 0,06 E 0~04 0,02 0,00 -0,02 -0,0d 0,12 0,10 0,08 _ O,06 E 0,04 O,02 O,OO -O,02 -O,04 O,10 O,08 O,06 O,04 0,02 0,00 -0,02 -0,04 0,08 0,06 0,04 E 0,02 0,00 -0,02 -0,04 -0,06 0,08 0,06 0,04 0,02 E 0,OO -0,02 -0,04 -O,06 -O,08 _ o _ 1N (i =' 0 5 10 15 20 25 30 35 40 ,^ x=24.04m o 1C 1 1 '` .? ` ~ 5 10 -~ ~E t 25 30 35 40 0 5 10 15 20 25 30 30 44 m _ 10 15 x = 37.04m ~'~ ~ f`\/ -qll ~ V~ 1 1 1 ~ 1 1 l l 5 10 15 20 25 40 20 25 30 35 40 Figure 11: Records of wave elevation at 6 different wave-probe locations for an initially harmonic wave of period 2.525 s and amplitude 0.029 m propagating over an uneven bottom (see Figure 12) as measured by Dingemans and calculated using BEShiWa (- - -) 13

Of 1 -0.2 -0.4 -0.6 -0.8 -1 \ 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 x [m] Figure 12: The two-dimensional bar-type bottom topography investigated by Dingemans REFERENCES Bolt, E.: "Fast ferry wash measurement and criteria," Proceedings of the FAST 2001, Southhampton, UK, 2001. Chen, X.-N. and Sharma, S.D.: "A slender ship moving at a near-critical speed in a shallow channel," Journal of Fluid Mechanics 291 (1995, S. 263-285. - Presented at the 18th Int. Congress of Theoretical and Applied Mech., Haifa, Israel, 1992. Chen, X.-N. and Uliczka, K.: "On ships in natural waterways," Proceedings of the RINA International Conference on Coastal Ships and Inland Waterways, London, 1999. Dingemans, M.W.: "Comparison of computations with Boussinesq-like models and laboratory measurements," MAST-G8M note, H1684, Delft Hydraulics, 1994. Dingemans, M.W.: "Water Wave Propagation over Uneven Bottoms", Advanced Series on Ocean Engineering, Vol. 13, 1997, Part II, pp. 635. Doctors, L.J., Philipps, S.J. and Day, A.H.: "Focussing the wave-wake system of a high-speed marine ferry," Pro- ceedinas ofthe FAST 2001, Southhampton, UK, 2001. Doyle, R., Whittaker, T.J.T. and Elsasser, B.: "A study of fast ferry wash in shallow water," Proceedings of the FAST 2001, Southhampton, UK, 2001. Feldtmann, M. and Garner, J.: "Seabed modifications to prevent wake wash from fast ferries," Proceedings of the RINA International Conference on Coastal Ships and Inland Waterways, London, 1999. Henn, R., Sharma, S. D. and Jiang, T. "Influence of Canal Topography on Ship Waves in Shallow Water," Proceed- ings of the 1 6th Int. Workshop on Water Waves and Floating Bodies, Hiroshima, Japan, 2001. Jiang, T.: "Ship Waves in Shallow Water," Fortschritt-Berichte VDI, Series 12, No. 466 with ISBN 3-18-346612-0, 2001. Jiang, T.: "Investigation of waves generated by ships in shallow water," Proceedings of the 22n~ Symposium On Naval Hydrodynamics Washington, D.C., USA, 1998. Koushan, K., Werenskiold, P., Zhao, R. and Lawless, J. "Experimental and theoretical investigation of wake wash," Proceedings ofthe FAST 2001, Southhampton, UK, 2001. MacLarlane, G.J. and Renilson, M.R.: "Wake wave - a rational method for assessment," Proceedings of the RINA International Conference on Coastal Ships and Inland Waterways London 1999. , , 14

Peregrine, D.H.: "Long waves on a beach," Journal of Fluid Mechanics, Vol. 27, 1967, pp. 815-827. Raven, H.C. "Numerical Wash Prediction Using A Free-Surface Panel Code," Proceedings of the RINA Interna- tional Conference on Hydrodynamics of High-Speed Craft - Wake Wash and Motion Control, London, 2000. Taylor, P.J.: "The Blockage Coefficient for Flow About an Arbitrary Body Immersed in a Channel," Journal of Ship Research Vol. 17, 1973, pp. 97-105. Yang, G.-Q., Faltinsen, O.M. and Zhao, R. Southhampton, UK, 2001. . . "Wash of ships in finite water depth," Proceedings of the FAST 2001, Zibell, H.G. and Grollius, W.: "Fast vessels on inland waterways," Proceedings of the RINA International Confer- ence on Coastal Ships and Inland Waterways, London, 1999. 15

DISCUSSION Stephane Cordier Bassin d'Essais des Carenes, France Ships inland waterway are confronted with changes in maneuvering behavior in shallow or restricted waters. Could you please tell us how this method can be used or extended to improve the prediction of maneuvering forces for ships in restricted water? AUTHORS' REPLY We thank Dr. Cordier for his question. For predicting the maneuvering forces on ships in restricted water, we need only to improve our approximation for the near-ship flow. Currently, we examine the possibility of coupling the BEShiWa program with different methods, such as with a panel program or a Euler solver or a RANSE solver. We hope to present our new results in the near future. DISCUSSION L.J. Doctors University of New South Wales, Australia I would like to express my appreciation to the three authors for a most interesting paper on the subject of wave generation, a matter of interest to many researchers who are aiming to reduce the potential damage done by high-speed ferries as well as traditional vessels, travelling near coastlines and river banks. The plots in Figure 1, in particular, are excellent for displaying the wave patterns created by the vessel at the various depth Froude numbers. It is encouraging, also, to observe the good comparison between the measured and calculated wave profiles in Figure 2. It is particularly impressive to see the calculations for the non- uniform bottom topography in Figure 3 and Figure 1 1. Referring specifically to Figure 2, could the authors comment on the likely relative accuracy of the BeShiWa (Boussinesq's Equations for Ship Waves) program, compared with, say, a more traditional linearized-free-surface method, in which no depth averaging is effected? That is, what sacrifice has been made in losing the details of the vertical distribution of the transverse velocities within the flow domain, in order to obtain the very impressive capabilities of BeShiWa? Secondly, can the authors verify that the effects of sinkage and trim are not included in their work? No doubt this would require a full near- field calculation (presumably not done here). The discusser feels that the effects of sinkage and trim are probably not important in most cases of practical interest. AUTHORS' REPLY We greatly appreciate Professor Doctors's comments and questions. The accuracy of predicting ship-wave propagation in shallow water by using the BEShiWa program is generally remarkable or at least practically acceptable in comparison with model tests. Till now, no attempt has been done by us to compare with a traditional linearized- free-surface method. A simple answer here would be that we do not have such a linear code. However, we would emphasize again our statement that due to the nonlinear and unsteady nature of ship waves in shallow water the linear theory remains to be a restricted approximation. Furthermore, it should be clarified that the vertical distribution of the transversal velocity components is explicitly described as an analytical function of the averaged horizontal velocity in the Boussinesq's shallow-water theory. So the vertical effects are not neglected, but analytically approximated in the BEShiWa program. Coming now to the second question, the effect of the sinkage and trim as well as the free surface elevation are simultaneously included in our near-field solution, see paragraph "Approximation of the Near-Ship Flow". As shown by Jiang (1998), the sinkage and trim could be well predicted by the BEShiWa program. The agreement of our calculations with model measurements was good not only in the subcritical speed range, but also in the transcritical and supercritical one. J

DISCUSSION H.C. Raven MARIN, The Netherlands This is an interesting paper on a topical subject. The extensive results illustrate the richness of wave phenomena occurring in practical situations; and show how strongly the particulars of the waterway determine which wave effects dominate and whether any wash problems will occur. In order to predict these phenomena, there is a need for a computational tool that incorporates the essential features and has reasonable efficiency. Boussinesq-type models seem to go a long way toward that objective, as the applications illustrate. My question is on the boundary condition at the ship hull; which is the one that generates the waves. In the present work, a 'slender-body' type condition is used: the passage of the ship imposes a lateral velocity distribution, which is averaged over the entire water depth. This is consistent with Boussinesq theory; but intuitively one would expect that this is less accurate for higher water depth / draught ratio's. Could the authors comment on their experience in this regard, and mention the water depth / draught ratio for the good results in Fig. 2? Secondly, is there a way to compute and incorporate the dynamic trim and sinkage in this method? AUTHORS' REPLY We thank Dr. Raven for his comments, particularly for his indication of our consistent approximation in using the Boussinesq's equations for the far-field flow and an extended slender-body theory for the near-ship flow. We agree with his presumption that our method is less accurate for higher ratios of water-depth to ship draught in the absolute sense of the increased water depth, but not in the relative sense of the ratio. For instance, the ratio for the good agreement in Figure 2 was approximately 4. The crucial parameter for using the BEShiWa program is the depth Froude number which should not be below the associated lower limit defined by Jiang (2001~. For the answer to the second question we refer to our reply to Professor Doctors on the previous page. DISCUSSION H. S. Choi Seoul National University, Korea In this paper, you have used the depth-averaged Boussinesq equations to describe wave field generated by ships moving on Fairways. Have you ever compared your numerical results with those obtained by FEM based on ON equations, which, for example, we presented at the 1 8eh SNH in Ann Arbor, 1990? AUTHORS' REPLY We thank Professor Choi for the reference of his work with the generalized Green-Naghdi (GN) equations. In comparison with the Boussinesq's equations the Green-Naghdi theory takes account of the fully nonlinear effects. As discussed by Jiang (2001), the application of the classical Boussinesq's equations for most practical cases is not limited by the nonlinear treatment but by the dispersion treatment. Various methods are derived in the work cited for the improvement of the dispersion relation of the Boussinesq's equations. Numerically we prefer the numerical more efficient Boussinesq approximation.

DISCUSSION H.C. Raven MARIN, The Netherlands This is an interesting paper on a topical subject. The extensive results illustrate the richness of wave phenomena occurring in practical situations; and show how strongly the particulars of the waterway determine which wave effects dominate and whether any wash problems will occur. In order to predict these phenomena, there is a need for a computational tool that incorporates the essential features and has reasonable efficiency. Boussinesq-type models seem to go a long way toward that objective, as the applications illustrate. My question is on the boundary condition at the ship hull; which is the one that generates the waves. In the present work, a 'slender-body' type condition is used: the passage of the ship imposes a lateral velocity distribution, which is averaged over the entire water depth. This is consistent with Boussinesq theory; but intuitively one would expect that this is less accurate for higher water depth / draught ratio's. Could the authors comment on their experience in this regard, and mention the water depth / draught ratio for the good results in Fig. 2? Secondly, is there a way to compute and incorporate the dynamic trim and sinkage in this method? AUTHORS' REPLY We thank Dr. Raven for his comments, particularly for his indication of our consistent approximation in using the Boussinesq's equations for the far-field flow and an extended slender-body theory for the near-ship flow. We agree with his presumption that our method is less accurate for higher ratios of water-depth to ship draught in the absolute sense of the increased water depth, but not in the relative sense of the ratio. For instance, the ratio for the good agreement in Figure 2 was approximately 4. The crucial parameter for using the BEShiWa program is the depth Froude number which should not be below the associated lower limit defined by Jiang (2001~. For the answer to the second question we refer to our reply to Professor Doctors on the previous page. DISCUSSION H. S. Choi Seoul National University, Korea In this paper, you have used the depth-averaged Boussinesq equations to describe wave field generated by ships moving on Fairways. Have you ever compared your numerical results with those obtained by FEM based on ON equations, which, for example, we presented at the 1 8eh SNH in Ann Arbor, 1990? AUTHORS' REPLY We thank Professor Choi for the reference of his work with the generalized Green-Naghdi (GN) equations. In comparison with the Boussinesq's equations the Green-Naghdi theory takes account of the fully nonlinear effects. As discussed by Jiang (2001), the application of the classical Boussinesq's equations for most practical cases is not limited by the nonlinear treatment but by the dispersion treatment. Various methods are derived in the work cited for the improvement of the dispersion relation of the Boussinesq's equations. Numerically we prefer the numerical more efficient Boussinesq approximation.

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